Determine the convergence or divergence of the following series. Fully justify your answer by referring to the test that you used. Show all necessary work! i) = COS ITT 900 ii) Σ=5 (-1)(1-1) nvn iii) XX=(1-3 iv) Σκ=2 3/k2 k4+4 V ke-* Zk=1
Answers
i) The series is divergent, ii) the series is convergent, iii) the series is convergent, iv) the series is convergent, and v) the series is convergent.
How to find the convergence or divergence of given series using various convergence tests?i) The series Σ cos(n π/2) is divergent. This is because the terms do not approach zero as n goes to infinity. In fact, the terms alternate between 1 and -1, so the series oscillates between the values 0 and 1, and does not converge.
ii) The series Σ (-1)^(n+1) / (n^n) is convergent. We can use the alternating series test to show this. The terms of the series alternate in sign and decrease in absolute value, since (n^n) grows faster than n. Additionally, lim(n→∞) 1/(n^n) = 0, so the series converges.
iii) The series Σ ((-1)^n * (n+1))/(3^n) is convergent. We can use the ratio test to show this:
|((-1)^(n+1) (n+2))/(3^(n+1))| / |((-1)^n (n+1))/(3^n)| = (n+2)/(3(n+1)) → 1/3 as n → ∞
Since the limit is less than 1, the series converges.
iv) The series Σ 3/(k^2 (k^4 + 4)) is convergent. We can use the comparison test with the p-series 1/(k^(5/2)) to show this:
0 < 3/(k^2 (k^4 + 4)) < 3/(k^(5/2))
Since Σ 1/(k^(5/2)) is a convergent p-series, Σ 3/(k^2 (k^4 + 4)) is also convergent.
v) The series Σ k*e^(-k) is convergent. We can use the ratio test to show this:
|a_(n+1)/a_n| = (n+1)e^(-n-1)/(ne^(-n)) = (n+1)/e
Since the limit of (n+1)/e as n goes to infinity is less than 1, the series converges.
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Related Questions
Determine whether each ordered pair is a solution or not a solution to this system of inequalities.
y< −x
2x+y>2
Answers
The ordered pair that is the solution of the given system of inequalities is (2, -2)
What is inequality?A relationship between two expressions or values that are not equal to each other is called inequality.
Given is a system of inequalities, y < -x and 2x+y > 2, we need to determine solution set of the given system of inequalities,
The inequalities are,
y < -x....(i)
2x+y > 2
y < 2-2x...(ii)
To find the ordered pair, put y = -x in equation Eq(ii) and replace < by =
-x = 2 - 2x
x = 2
y = -2
Therefore, the ordered pair, is (2, -2) {look at the graph attached}
Hence, the ordered pair that is the solution of the given system of inequalities is (2, -2)
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Can someone pls explain what I’m supposed to do I don’t really understand
Answers
Answer:
There are only 3 differences perform
Step-by-step explanation:
I. If x is negative, y is negative then the answer is possitive
eg. x is -2, y is -3
-2 × -3 = 6
II. If x is negative, y is positive then the answer is negative
eg. x is -1, y is 2
-1 × 2 = -2
III. If x is possitive, y is possitive then the answer is positive
eg. x is 2, y is 5
2 × 5 = 10
Alt. No matter how long x, y, z, a, b there always apply only 3 rules above. Multiply or Devide are the same rule.
Ask me for anything :D
8) \( f(x)=x^{4}-24 x^{2} \) FIND THE INTERVALS WHERE \( f \) is CU, CD. (IDENTIFT)
Answers
The function \(f(x) = x^4 - 24x^2\) is increasing on the intervals \((-∞, -2)\) and \((2, ∞)\), and decreasing on the interval \((-2, 2)\).
To determine where the function \(f(x) = x^4 - 24x^2\) is increasing or decreasing, we need to find its critical points and analyze the intervals between them.
First, let's find the derivative of \(f(x)\) using the power rule: \(f'(x) = 4x^3 - 48x\).
To find the critical points, we set \(f'(x) = 0\) and solve for \(x\):
\(4x^3 - 48x = 0\).
Factoring out 4x, we get: \(4x(x^2 - 12) = 0\).
This equation has three solutions: \(x = 0, x = -2\), and \(x = 2\).
Next, we create a sign chart to analyze the intervals between these critical points.
On the interval \((-∞, -2)\), we can test a value less than -2, such as -3. Plugging it into \(f'(x)\), we get a positive result, indicating that \(f(x)\) is increasing in this interval.
On the interval \((-2, 2)\), we can test a value between -2 and 2, such as 0. Plugging it into \(f'(x)\), we get a negative result, indicating that \(f(x)\) is decreasing in this interval.
On the interval \((2, ∞)\), we can test a value greater than 2, such as 3. Plugging it into \(f'(x)\), we get a positive result, indicating that \(f(x)\) is increasing in this interval.
Therefore, the function \(f(x) = x^4 - 24x^2\) is increasing on the intervals \((-∞, -2)\) and \((2, ∞)\), and decreasing on the interval \((-2, 2)\).
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What are the roots and the sum and products of those roots of 4x^2-12x+15=0
Answers
PLEASE GIVE BRAINLIEST
i hope this helps, thank you and have a good day ;)
To find the roots of the quadratic equation 4x^2-12x+15=0, we can use the quadratic formula:x = (-b ± √(b^2-4ac)) / 2a
where a = 4, b = -12, and c = 15.Substituting these values into the formula, we get:x = (-(-12) ± √((-12)^2 - 4(4)(15))) / 2(4)
x = (12 ± √(144 - 240)) / 8
x = (12 ± √(-96)) / 8 Since the square root of a negative number is not a real number, this equation has no real roots.
Instead, the roots are complex numbers of the form a ± bi, where i is the imaginary unit (√-1).
The sum of the roots of a quadratic equation ax^2+bx+c=0 is given by -b/a, and the product of the roots is given by c/a.
In this case, we have a=4, b=-12, and c=15, so the sum of the roots is:
sum of roots = -b/a = -(-12)/4 = 3
and the product of the roots is:
product of roots = c/a = 15/4
Therefore, the roots are complex numbers, and their sum is 3 and their product is 15/4.
Mrs Rani is 36 years old and her daughter is 4 years old. In how many years’ time
will she be three times her daughter’s age?
Answers
Answer:
MRS RANI 39 AND DAGHTER 7
PLEASE HELP 20 POINTS‼️‼️‼️
Answers
Answer:
x = 0, x = - 1
Step-by-step explanation:
The solutions are at the points of intersection of the graphs of f(x) and g(x)
The points of intersection occur at x = 0 and x = - 1
Then the solutions are x = 0, x = - 1
Please look at the picture, I need help ASAP.
Answers
See below for the proof that the areas of the lune and the isosceles triangle are equal
How to prove the areas?The area of the isosceles triangle is:
\(A_1 = \frac 12r^2\sin(\theta)\)
Where r represents the radius.
From the figure, we have:
\(\theta = 90\)
So, the equation becomes
\(A_1 = \frac 12r^2\sin(90)\)
Evaluate
\(A_1 = \frac 12r^2\)
Next, we calculate the length (L) of the chord as follows:
\(\sin(45) = \frac{\frac 12L}{r}\)
Multiply both sides by r
\(r\sin(45) = \frac 12L\)
Multiply by 2
\(L = 2r\sin(45)\)
This gives
\(L = 2r\times \frac{\sqrt 2}{2}\)
\(L = r\sqrt 2\)
The area of the semicircle is then calculated as:
\(A_2 = \frac 12 \pi (\frac{L}{2})^2\)
This gives
\(A_2 = \frac 12 \pi (\frac{r\sqrt 2}{2})^2\)
Evaluate the square
\(A_2 = \frac 12 \pi (\frac{2r^2}{4})\)
Divide
\(A_2 = \frac{\pi r^2}{4}\)
Next, calculate the area of the chord using
\(A_3 = \frac 12r^2(\theta - \sin(\theta))\)
Recall that:
\(\theta = 90\)
Convert to radians
\(\theta = \frac{\pi}{2}\)
So, we have:
\(A_3 = \frac 12r^2(\frac{\pi}{2} - \sin(\frac{\pi}{2}))\)
This gives
\(A_3 = \frac 12r^2(\frac{\pi}{2} - 1)\)
The area of the lune is then calculated as:
\(A = A_2 - A_3\)
This gives
\(A = \frac{\pi r^2}{4} - \frac 12r^2(\frac{\pi}{2} - 1)\)
Expand
\(A = \frac{\pi r^2}{4} - \frac{\pi r^2}{4} + \frac 12r^2\)
Evaluate the difference
\(A = \frac 12r^2\)
Recall that the area of the isosceles triangle is
\(A_1 = \frac 12r^2\)
By comparison, we have:
\(A = A_1 = \frac 12r^2\)
This means that the areas of the lune and the isosceles triangle are equal
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There are n counters in a bag.
8 of the counters are red and the rest are blue. Adam takes a counter from the bag at random and does not replace it.
He then takes another counter at random from the bag.
The probability that Adam takes two blue counters is 1/5
(a) Show that n² - 21n +90=0
Answers
Answer:
Let's start by using the fact that the probability of getting two blue counters is 1/5.
The probability of getting a blue counter on the first draw is (n-8)/n.
After taking out one blue counter, the probability of getting another blue counter is (n-9)/(n-1).
So the probability of getting two blue counters is:
(n-8)/n * (n-9)/(n-1) = 1/5
Multiplying both sides by 5n(n-1), we get:
5(n-8)(n-9) = n(n-1)
Expanding and simplifying, we get:
5n² - 85n + 360 = n² - n
Rearranging, we get:
n² - 21n + 90 = 0
Therefore, n² - 21n + 90 = 0, which is the desired result.
Step-by-step explanation:
-35÷6=
-350÷6-350-6=
350 - 60 =
Answers
Answer:
The first one is 5.83 or -5 5/6
The second one is -414 1/3
The last one is 290
Angles 1 and 2 are supplementary. 2 lines intersect to form angles 1 and 2. Which equation represents the relationship between their measures?
Answers
Answer:
\(\angle 1 + \angle 2 = 180^o\)
Step-by-step explanation:
Given
\(\angle 1\) and \(\angle 2\)
Required
The relationship between them \(\angle 1\) and \(\angle 2\)
From the question, we understand that \(\angle 1\) and \(\angle 2\) are supplementary
Supplementary angles add up to 180.
So, the relationship between \(\angle 1\) and \(\angle 2\) is:
\(\angle 1 + \angle 2 = 180^o\)
Find the amount accumulated after
investing a principal P for t years at an
interest rate compounded annually.
P = $15,500
r = 9.5%
t = 12
Hint: A = P (1 + ) kt
A = $[?]
Round your answer to the nearest cent (hundredth).
Answers
The amount accumulated after investing a principal P for t years at an interest rate compounded annually is $46,057.58.
How to solve compound interest ?Compound interest is the interest you earn on interest. Compound interest is the interest calculated on the principal and the interest accumulated over the previous period.
Therefore, let's find the amount accumulated after investing a principal P for t years at an interest rate compounded annually.
\(A = p(1 + \frac{r}{n} )^{nt}\)
where
P = principalr = ratet = timen = number of timep = 15,500
r = 9.5%
t = 12
n = 1
\(A = 15500(1 + \frac{0.095}{1} )^{1(12)}\)
A = 15,500.00(1 + 0.095)¹²
A = $46,057.58
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$46,057.58, answer for acellus
The solutions to a system of linear inequalities are located in the region where all the shaded regions of the inequalities overlap. What is the region of overlap called?
A. vertex
B. boundary line
C. intersection
D. test point
Answers
All the shaded regions of the overlap inequalities represents the located solution region of the system of linear inequalities , the overlap region is called intersection ( option C ).
Graph is attached.
Let us consider the system of linear inequalities are x + y > 1 and x -y <1.Represents both the system of inequality on the graph.Plot x + y > 1, region above x >1, y > 1 represents the inequality.Plot x - y < 1 region x < 1 , y > -1 represents the inequality.Shaded region of both the inequalities intersect each other represents the solution region of the system of inequalities.Solution region is the shaded region which get overlap.Graph is attached.
Therefore, the solution region is represented by the shaded region called (option C) intersection area of the system of linear inequalities.
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a die is tossed 180 times with the following results: x 1 2 3 4 5 6 f 28 36 36 30 27 23 is this a balanced die? use a 0.01 level of significance
Answers
Based on the chi-square test, there is no significant evidence to suggest that the die is not balanced.
We have,
To determine if the die is balanced, we can perform a chi-square test of goodness of fit.
The null hypothesis is that the die is balanced, and the alternative hypothesis is that the die is not balanced.
First, let's calculate the expected frequencies for each outcome assuming the die is balanced. Since there are 180 tosses in total, each outcome is expected to have an equal probability of 1/6.
Expected frequency for each outcome
= (Total tosses) x (Probability of each outcome)
Expected frequency for each outcome = (180) x (1/6)
Expected frequency for each outcome = 30
Now, we can calculate the chi-square test statistic using the formula:
χ² = Σ [(Observed frequency - Expected frequency)² / Expected frequency]
Let's calculate the chi-square test statistic:
χ² = [(28 - 30)² / 30] + [(36 - 30)² / 30] + [(36 - 30)² / 30] + [(30 - 30)² / 30] + [(27 - 30)² / 30] + [(23 - 30)² / 30]
χ² = [(-2)² / 30] + [(6)² / 30] + [(6)² / 30] + [(0)² / 30] + [(-3)² / 30] + [(7)² / 30]
χ² = 4/30 + 36/30 + 36/30 + 0/30 + 9/30 + 49/30
χ² = 134/30
χ² ≈ 4.467
Next, we need to compare the calculated chi-square value to the critical chi-square value at a significance level of 0.01 and degrees of freedom equal to the number of outcomes minus 1 (6 - 1 = 5).
Looking up the critical chi-square value in a chi-square distribution table with 5 degrees of freedom and a significance level of 0.01, we find it to be approximately 15.086.
Since the calculated chi-square value (4.467) is less than the critical chi-square value (15.086), we fail to reject the null hypothesis.
Therefore, we do not have sufficient evidence to conclude that the die is not balanced at a 0.01 level of significance.
Thus,
Based on the chi-square test, there is no significant evidence to suggest that the die is not balanced.
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for an arbitrary collection of closed intervals, will the intersection of all of those intervals always be a closed interval if it is nonempty?
Answers
The intersection of all of those intervals always be a closed interval if it is nonempty
Given that
for an arbitrary collection of closed intervals
the intersection of all of those intervals always be a closed interval if it is nonempty it's true
An arbitrary (finite, countable, or uncountable) intersection of
closed sets is closed.
Let C0 = [0, 1], C1 = [0, 1/3] ∪ [2/3, 1], and
C2 = [0, 1/9] ∪ [2/9, 1/3]∪ [2/3, 7/9] ∪ [8/9, 1]
In general, let
Cn =(1/3 \(C_{n-1}\))U(1/3\(C_{n-1}\) + 2/3)
Let C = ∩n>1Cn, called Cantor’s middle-third set. Show that C is
a closed set.
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HELP ME QUICK!!!
The table models how the population of a city has changed over time. What does the y-intercept represent?
Answers
Answer:
The y intercept represents the population of the city in the year of 1985.
Step-by-step explanation:
Since it is 0 years after 1985, it means that the population was 122,000 in 1985.
Stacy bought 5 tickets to a baseball game for her family. The first two tickets cost $23 each. The second two tickets cost $28 each. Find the cost of the fifth ticket if the average price Stacy paid for the five tickets was $27.
Answers
Answer:
33
Step-by-step explanation:
23+23+28+28+33=135
$135/5tickets=$27avg
The cost of the fifth ticket is $33.
Stacy bought two tickets at $23 each, so the total cost for the first two tickets is 2×23.
She bought two tickets at $28 each, so the total cost for the second two tickets is 2×28.
Adding these costs together, we have:
2×23+2×28+x=46+56+x=102+x.
The average price Stacy paid for the five tickets was given as $27. We can set up an equation using the average:
46+56+x/5 = 27
Now, we can solve for x:
102+x=5×27,
x=135−102
x=33.
So, the cost of the fifth ticket is $33.
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i need some help again please
Answers
Answer:
Step-by-step explanation:
d = 96 inches
\(T_{max}=\dfrac{2*96}{3}+5\\\\\\=2*32 + 5\\\\= 64 + 5 \\\\T_{max} =69 \ inches\\\\\\\\T_{min}=\dfrac{96}{3}-2\\\\\\=32-2\\\\T_{min}= 30 \ inches\)
Answer:
The answer is, cool balls 420
When comparing three or more populations means within a set of quantitative data that is categorized according to one factor/treatment, a one-way ANOVA is appropriate.a. It is also appropriate in this situation, however, to compare two means at a time using multiple independent two sample t-tests. b. It is appropriate to compare two means at a time with independent two sample t-tests but it might be time-consuming.c. It is not appropriate to compare two means at a time in the way described. This would inflate the overall Type I Error and is a 'Multiple Testing' problem. The one-way ANOVA controls for the Type I Error and should be used instead.
Answers
According one-way ANOVA, the test is false.
We learn about one-way ANOVA
"One-Way ANOVA, also known as "analysis of variance," examines the refers to two or more independent groups to see if there is statistical support for the notion that the related population means are statistically substantially different."
According to the given information, it is inappropriate to compare two means at a time using multiple independent two sample t-tests. It will create multiple testing problem and error.
So using one way ANOVA test when comparing three or more populations refers to within a set of quantitative data that is categorized according to one factor/treatment and compare two means at a time using multiple independent two sample t-tests is false.
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Jaylen used the seat number for each of the 6,500 fans' seats in the stands at a college football game and put the numbers in a computer program. He randomly chose 200 numbers and found that 36 of those people had also purchased a parking voucher. Estimate the number of fans in the stands at a sold-out football game that purchased a parking voucher. Explain.
Answers
Answer:
1170 seats
Step-by-step explanation:
Given
\(Seats = 6500\)
From the computer program, we have:
\(Sample = 200\)
\(Voucher = 36\)
First, we calculate the proportion (p) of those that bought the voucher
\(Proportion = \frac{Voucher}{Sample}\)
\(p = \frac{36}{200}\)
\(p = 0.18\)
Using the larger population (i.e. 6500 seats), the esitimate of people that have bought the voucher is then calculated as:
\(Estimate= Proportion * Seats\)
\(Estimate= 0.18 * 6500\)
\(Estimate= 1170\)
A balalaika is a Russian stringed instrument. Show that the triangular parts of the two balalaikas are congruent for x = 6.
Answers
*the diagram of the Russian stringed instrument is attached below.
Answer/Step-by-step explanation:
To show that the traingular parts of the two balalaikas instruments are congruent, substitute x = 6, to find the missing measurements that is given in both ∆s.
Parts of the first ∆:
WY = (2x - 2) in = 2(6) - 2 = 12 - 2 = 10 in
m<Y = 9x = 9(6) = 54°.
XY = 12 in
Parts of the second ∆:
m<F = 72°
HG = (x + 6) in = 6 + 6 = 12 in
HF = 10 in
m<G = 54°
m<H = 180 - (72° + 54°)
m<H = 180 - 126
m<H = 54°
From the information we have, let's match the parts that are congruent to each other in both ∆s:
WY ≅ FH (both are 10 in)
XY ≅ GH (both are 12 in)
<Y ≅ <G (both are 54°)
Thus, since two sides (WY and XY) and an included angle (<Y) of ∆WXY is congruent to two corresponding sides (FH and GH) and an included angle (<G) in ∆FGH, therefore, ∆WXY ≅ ∆FGH by the Side-Angle-Side (SAS) Congruence Theorem.
This is enough proof to show that the triangular parts of the two balalaikas are congruent for x = 6.
I don't understand.......
Answers
Answer:
Step-by-step explanation:
the dot is on the 32 and the line is the 3rd line
Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p.(a) the probability of no successes(b) the probability of at least one success(c) the probability of at most one success(d) the probability of at least two successes(e) the probability of no failures(f) the probability of at least one failure(g) the probability of at most one failure(h) the probability of at least two failures
Answers
The probability of at least two failures is 1 minus the probability of 0 or 1 failure, which is 1 - [p^n + nqp^(n-1)].
The probability of a success in one Bernoulli trial is given by p, and the probability of a failure is q = 1 - p.
(a) The probability of no successes is (1-p)^n.
(b) The probability of at least one success is 1 minus the probability of no successes, which is 1 - (1-p)^n.
(c) The probability of at most one success is the sum of the probabilities of 0 and 1 successes, which is (1-p)^n + np(1-p)^(n-1).
(d) The probability of at least two successes is 1 minus the probability of 0 or 1 success, which is 1 - [(1-p)^n + np(1-p)^(n-1)].
(e) The probability of no failures is the same as the probability of n successes, which is p^n.
(f) The probability of at least one failure is 1 minus the probability of no failures, which is 1 - p^n.
(g) The probability of at most one failure is the sum of the probabilities of 0 and 1 failures, which is p^n + nqp^(n-1).
(h) The probability of at least two failures is 1 minus the probability of 0 or 1 failure, which is 1 - [p^n + nqp^(n-1)].
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WILL MARK BRAINLIEST PLSEASE HELP!!!!
Answers
Answer:
Slope is 3/5
Step-by-step explanation:
help:(!!.!.!.!.!.!!.!.
Answers
Answer:
Step-by-step explanation:
10(5)=50
Answer:
50
Step-by-step explanation:
A=bh
50 = 10 x 5
Using the SAS Congruence Theorem
Try it
Angles Segmenis Triangles Statements Reasons
Given: UK || LMJKLM
L is the midpoint of JN.
Prove: AJLKALNM
ZLJK
ZNIM
VA
J
K
Statements
Reasons
L
M
N
Assemble the proof by dragging tiles to
the Statements and Reasons columns.
Answers
Sorry this got to you super late. At least others looking for the answer can find it.
The proof by S A S congruence rule is given below.
Given from fig, JK || LM.
And JK = LM
Since JK is parallel to LM and JN is a transversal so the angle made on the same side of JN are corresponding angles which are equal to each other.
So, angle KJL = angle MLN........equation 1
Also given L is the midpoint of JN so, JL = NL.....equation 2
Now, in triangle JLK and triangle LNM
JK = LM (given)
angle KJL = angle MLN (from 1)
JL = NL (from 2)
So by S A S congruence rule, triangle JLK conguent triangle LNM proved.
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solve this question fast please solve this fast
Answers
Answer:
15%
Step-by-step explanation:
RS. 6450 = 15%
6450 percent *15 =
(6450:100)*15 =
(6450*15):100 =
96750:100 = 967.5
Answer:
monthly income=rs14000
annual income =rs14000×12=rs 168000
since it is greater than rs125000
he needed to pay tax=x% let
tax amount =6450
x% of rs (168000-125000)=6450
x/100×43000=6450
x=6450/430=15
therefore required tax %=15%
jared needs to drill holes 2 1/4 inches apart in a 36 inch section of his garage. which expression can he use to figure out the number of holes he will need to drill
Answers
Answer: 36 / 2 1/4
Step-by-step explanation:
From the question, we are informed that Jared needs to drill holes 2 1/4 inches apart in a 36 inch section of his garage. The expression that he can use to figure out the number of holes he will need to drill will be:
= 36 ÷ 2 1/4
From the expression we can then calculate the number of hours to drill. This will be:
= 36 / 2 1/4
= 36 / 9/4
= 36 × 4/9
= 16
He'll need to drill 16 holes
Answer:
18c
= 18(36)
= 648
Step-by-step explanation:
i hope this helps and im on edmentum
pls help 10 minutes left to submit everything is on the image below
Answers
Answer:
a. 5
b. 1
c. 132
d. -2
Step-by-step explanation:
a. The square root of 4 is 2, the absolute value of -3 is 3. 2 + 3 = 5
b. 4 - (-3) / -7 = 7/ -7 = 1
c. 2 * 4^3 = 128 + (-3) - (-7) = 132
d. 2 (-7 + 3) = -8, square root to the third power of -8 is -2
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if a circle has a radius of 7/4 cm, what is the diameter?
Answers
Answer:
The diameter of the circle is 7/2.
Step-by-step explanation:
⭐What is the radius of a circle?
the distance from a point on the circumference of the circle to the center of the circle⭐What is the diameter of a circle?
the distance from a point on the circumference of the circle to another point on the circumference of the circlethe diameter must cut through the center of the circlethe diameter is double the amount of the radiusThus, the diameter of said circle is twice the radius of said circle.
diameter = 2(radius)
diameter = 2(7/4) . . . . substitute known values
diameter = 14/4 . . . . multiply the numerator & denominator
diameter = 7/2 . . . . simplify
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factor m^4 + 9m^2 + 20 completely
Answers
Answer:
Step-by-step explanation:
m^2(m^2+9)+20